The factors of a number are all positive integers that divide evenly into the number. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

To multiply terms with radicals, multiply the coefficients and radicands separately:

8\( \sqrt{3} \) x 4\( \sqrt{9} \)
(8 x 4)\( \sqrt{3 \times 9} \)
32\( \sqrt{27} \)

Now we need to simplify the radical:

32\( \sqrt{27} \)
32\( \sqrt{3 \times 9} \)
32\( \sqrt{3 \times 3^2} \)
(32)(3)\( \sqrt{3} \)
96\( \sqrt{3} \)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 5}{2 x 5} \) + \( \frac{8 x 1}{10 x 1} \)

\( \frac{20}{10} \) + \( \frac{8}{10} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{20 + 8}{10} \) = \( \frac{28}{10} \) = 2\(\frac{4}{5}\)

First, solve for \( \left|a + 7\right| \):

\( \left|a + 7\right| \) + 1 = 3
\( \left|a + 7\right| \) = 3 - 1
\( \left|a + 7\right| \) = 2

The value inside the absolute value brackets can be either positive or negative so (a + 7) must equal + 2 or -2 for \( \left|a + 7\right| \) to equal 2:

So, a = -9 or a = -5.

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]. They share 4 factors [1, 2, 4, 8] making 8 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{48} \) = \( \frac{\frac{40}{8}}{\frac{48}{8}} \) = \( \frac{5}{6} \)